This is exactly why reputable RPT operations will have an obstacle clearance procedure or OCP and a procedure to follow in the event that you've already commenced the turn on a SID. In any case it might well be different to the SID.
Unless conducting certain types of RNP approaches, it is also what you would follow in the event of a OEI go around from below the minima in IMC.
My current airline and the previous published both an OCP and a 'post procedure'
For example. EFATO from YSSY 34r, if you've already started the turn then track 090 and climb to the MSA. If you haven't yet stared the turn then track straight ahead till 10nm then turn right and track 090 climbing to the MSA.
It's written our TO card and checked during the pre flight.
In short, if one engine keeps going, then you've always got an option.